"h-topology and applications"

These lectures will introduce Voevodsky's \(h\)-topology and some variants. Our focus will be on geometric applications (such as Hodge theory and singularities)

"Miscellaneous on rational points and fundamental groups"

The guiding conjecture of the lectures is the \(C_1\) conjecture, due to Lang-Manin-Kollár, predicting that a rationally connected variety over a \(C_1\) field has a rational point (with some separability assumption in positive characteristic). The essential case where it is still unknown is over the maximal unramified extension of \(\mathbb{Q}_p\). We will discuss various aspects of the problem, and related questions.

"The geometry of hyperkähler varieties"

Definition and examples of compact hyperkähler manifolds; General results on compact hyperkähler manifolds: deformations, period map, B-B quadratic form, Kähler cone, Torelli Theorems. Moduli of polarized hyperkähler varieties. Explicit examples of locally complete families of polarized hyperkähler varieties. Chow ring of hyperkähler varieties. Preliminary notes can be found here and here.